3-Manifolds and Applications to Geometry

3-流形及其在几何中的应用

• 批准号: 0083097
• 负责人: Walter Neumann
• 金额: $ 5.1万
• 依托单位: Barnard College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0083097&HistoricalAwards=false
• 关键词: Manifolds; Applications; Geometry

Proposal: DMS-0083097PI: Walter D. NeumannDespite the venerable history of the algebraic geometry, many difficultproblems remain even in low dimensions. Methods of low dimensionaltopology have proved very helpful here, and part of this project concernsa continuation of the investigator's successful application of suchmethods to the study of singularities of complex surfaces and the topologyof plane affine curves. One issue to be addressed is explicit analyticdescription of singularities with given topology. The general question isnotoriously difficult, but the investigator and J. Wahl have recently madea series of conjectures on realising rationally Gorenstein singularitieswith rational homology sphere links as explicit quotients of completeintersections and are making significant progress towards theirresolution. 3-manifold topology is also being applied to the study of theglobal topology and analytic classification issues for two-variablepolynomials. These issues are of intrinsic interest as well as havingpotential to contribute to the resolution of the famous JacobianConjecture. In a rather different direction, the investigator will studyinvariants of 3-dimensional manifolds that relate to number theory andalgebra, namely the so-called character and eigenvalue varieties and theBloch invariant. Connections are emerging between these invariants thatwill inform both topology and algebra/number theory.Algebraic Geometry, which is essentially the study of the zero sets offamilies of polynomials, has always been an important area offundamental research, and is also important in such diverseapplications as control theory and communications. The project willapply topological methods to algebraic geometry, to obtain results onhow to realise and classify families of polynomials that lead toparticular topologies. The methods of 3-dimensional topology are ofparticular importance to this project, and there is also feedback fromalgebra and number-theory to topology, as well as some interactionwith theoretical physics and cosmology. The project also studiesnew insights that these connections bring to all these fields.

提案：DMS-0083097PI：Walter D. Neumann 尽管代数几何有着悠久的历史，但即使在低维情况下，许多难题仍然存在。低维拓扑方法在这里已被证明非常有用，该项目的一部分涉及研究者将此类方法成功应用于复杂曲面奇点和平面仿射曲线拓扑研究的延续。要解决的一个问题是给定拓扑的奇点的显式分析描述。这个一般性问题是出了名的困难，但研究者和 J. Wahl 最近提出了一系列猜想，以理性同调球面链接作为完全交集的显商来实现理性 Gorenstein 奇点，并在解决这些问题方面取得了重大进展。三流形拓扑也被应用于二元多项式的全局拓扑和解析分类问题的研究。这些问题具有内在意义，并且有可能有助于解决著名的雅可比猜想。 在一个相当不同的方向上，研究者将研究与数论和代数相关的 3 维流形的不变量，即所谓的特征和特征值簇以及布洛赫不变量。 这些不变量之间正在出现的联系将为拓扑和代数/数论提供信息。代数几何本质上是对多项式零集族的研究，一直是基础研究的一个重要领域，并且在控制理论和通信等多种应用中也很重要。该项目将拓扑方法应用于代数几何，以获得如何实现和分类导致特定拓扑的多项式族的结果。 3维拓扑的方法对于这个项目特别重要，而且还有代数和数论对拓扑的反馈，以及与理论物理和宇宙学的一些相互作用。该项目还研究这些联系给所有这些领域带来的新见解。